Question

1) If the point\(\left( {{\bf{5,3}}} \right)\)is on the graph of an even function, what other point must also be on the graph?

(2) If the point\(\left( {{\bf{5,3}}} \right)\)is on the graph of an odd function, what other point must also be on the graph?

Short answer

(1) The other point on the graph for even function must be\(\left( {{\bf{ - 5,3}}} \right)\).

(2) The other point on the graph for the odd function must be\(\left( {{\bf{ - 5, - 3}}} \right)\).

Step-by-step solution

Determination of another point on the graph

(1).

The graph of the even function is always symmetric about the y-axis. Therefore, the y coordinate of the point remains the same and the sign of the x coordinate of the point becomes opposite.

The y coordinate of a given point (5,3) is 3 which remains the same for other points and only the x coordinate of point (5,3) becomes opposite, so the x coordinate of the other point will be\( - 5\). Thus, the coordinates of the other point will be\(\left( { - 5,3} \right)\).

Therefore, the other point on the graph must be\(\left( { - 5,3} \right)\).

Determination of another point on the graph

(2).

The graph of an odd function is always symmetric with respect to the origin. So the coordinates of a point in one quadrant become opposite to that point in the opposite quadrant.

The x and y coordinate of the given point is 5 and 3 in the first quadrant which becomes -5 and -3 in the third quadrant. Thus the other point on the graph of the odd function will be\(\left( { - 5, - 3} \right)\).

Therefore, the other point on the graph must be\(\left( { - 5, - 3} \right)\).